if #v=1148sqrt(p)# where #p# is a function of #t# then find #(dv)/dt# when #p=44.0# and #dp/dt=0.307#?

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Answer 1 · 2017-07-21T14:10:52

# [(dv)/(dt)]_(p=44.0) = 26.3 \ ft (s^(-2))# to three significant figures

We have:

# v = 1148sqrt(p) #
# \ \ = 1148p^(1/2) #

Differentiating wrt #p# we get:

# (dv)/(dp) = 1/2(1148)p^(-1/2) #
# " " = 574/sqrt(p) #

We are also give that:

# (dp)/dt =0.307 #

By the chain rule we have:

# (dv)/(dt) = (dv)/(dp) * (dp)/(dt) #
# " " = 574/sqrt(p) * 0.307#
# " " = 176.218/sqrt(p)#

So when #p=44.0# we have:

# [(dv)/(dt)]_(p=44.0) = 176.218/sqrt(44) #
# " " = 26.565863 ... ft \ s^(-2)#